Numerical construction of initial data sets of binary black hole type using a parabolic-hyperbolic formulation of the vacuum constraint equations

Abstract

In this paper we investigate the parabolic-hyperbolic formulation of the vacuum constraint equations introduced by R{á}cz with a view to constructing multiple black hole initial data sets without spin. In order to respect the natural properties of this configuration, we foliate the spatial domain with 2-spheres. It is then a consequence of these equations that they must be solved as an initial value problem evolving outwards towards spacelike infinity. Choosing the free data and the "strong field boundary conditions" for these equations in a way which mimics asymptotically flat and asymptotically spherical binary black hole initial data sets, our focus in this paper is on the analysis of the asymptotics of the solutions. In agreement with our earlier results, our combination of analytical and numerical tools reveals that these solutions are in general not asymptotically flat, but have a cone geometry instead. In order to remedy this and approximate asymptotically Euclidean data sets, we then propose and test an iterative numerical scheme.

arXiv comment36 pages, 8 figures. Matches the published version. Title and abstract changed slightly. Introduction and discussion rewritten. Several typos fixed. No change of the results or conclusions

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Cited by in corpus (6)

• Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints
• Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations
• Explorations of the infinite regions of space-time
• Asymptotically hyperboloidal initial data sets from a parabolic-hyperbolic formulation of the Einstein vacuum constraints
• Is it possible to construct asymptotically flat initial data using the evolutionary forms of the constraints?
• Black hole initial data by numerical integration of the parabolic-hyperbolic form of the constraints